# Finding a superbase in a lattice of Voronoi first kind

An $$n$$-dimensional lattice in $$\mathbb R^n$$ is said to be of Voronoi’s first kind if it there exists $$n+1$$ vectors $$b_1,\cdots b_{n+1}$$ (called the superbase) such that

1. $$\{b_1,\ldots,b_n \}$$ is a basis,
2. $$b_1+\cdots +b_{n+1}=0$$,
3. $$q_{ij}=b_i^Tb_j\le 0$$ for all $$i\ne j$$. The $$q_{ij}$$ are called selling parameters.

Condition (2) is called the superbase condition and condition (3) the obtuse condition.

Given a lattice which is known to be Voronoi's first kind, are any methods known to find a superbase for it?